Saying “what is today’s day?” sounds awkward and is not really used. There is nothing wrong with saying “What day is it today?”, which would mean that someone is asking what day of the week it is today for example, Monday or Saturday.
"To this day" is an unusual phrasing, only used to talk about something that has been going on a long time and is still that way. It would not be used for short time periods or indicating the turn-over of a calendar day in the minute past midnight. You should generally think of not using it for anything much less than a year; and usually much longer spans of time.
For instance, a difficult problem known to the ancient Greeks might turn out to be "still unsolved, to this day." And to put stronger and more mysterious emphasis on how odd it is that no one has figured it out, you might phrase it as "still unsolved, even to this very day".
Going for something less dramatic, the casual "until today" doesn't specify a clock time. If I said "I didn't know that until today", I am not suggesting I got an email about it at midnight. I am saying "I didn't know this fact prior to the current calendar day, and only found out at a clock time prior to us speaking now."
(Note: We sometimes consider a very late hour like 1:00 AM to be the previous day's "night". Hence I might say "I saw them yesterday night" on Monday, even if I had actually seen them earlier on that same Monday at 1:00 AM.)
To get more specific about "day" you have to modify it. There are phrases like "You will receive it by the next business day" which means that if someone says that on a Friday, you will not get it until Monday (assuming Monday is not a work holiday). "The next day, she opened the letter" or similar phrases are ambiguous about time.
You will have to be specific if a specific time is meant. For instance: "As the clock struck midnight, the phantom appeared."
That is all very far from the understanding when people would use "to this day". Once again, it is only a rare phrasing used to indicate a long time frame extending all the way up to the present...usually for dramatic and/or mysterious effect.
Best Answer
1 and 3 are both correct. 2 is not.
1 is a complete, grammatically correct sentence.
3 is not a complete sentence, but in common speech and informal writing we often do not use complete sentences when the meaning is clear. This is especially true for questions. If someone asks, say, "What color shirt is Bob wearing today?", most people would answer simply "Red" (or whatever color, of course). It's not a complete sentence, but we know that you mean "Bob is wearing a red shirt today".
2 is not correct because there is nothing that is "in" the morning. If you asked, "When is the staff meeting?", then it would make sense for me to say, "The staff meeting is in the morning", or simply "In the morning." But if you ask what time of day it is, there is nothing "in" this time. It just is.